SlabLU: A Sparse Direct Solver for Elliptic PDEs on Rectangular Domains
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The paper describes a sparse direct solver for the linear systems that arise from the discretization of an elliptic PDE on a rectangular domains. The scheme decomposes the domain into thin subdomains, or “slabs”. Within each slab, a local factorization is executed that exploits the geometry of the local domain. A global factorization is then obtained through the LU factorization of a block-tridiagonal reduced system. The general two-level framework is easier to implement and optimize for modern latency-bound architectures than traditional multi-frontal schemes based on hierarchical nested dissection orderings. The solver has complexity O(N^5/3) for the factorization step, and O(N^7/6) for each solve once the factorization is completed. The solver described is compatible with a range of different local discretizations, and numerical experiments demonstrate its high performance for standard finite difference discretizations on a regular grid. The technique becomes particularly efficient when combined with very high-order convergent multi-domain spectral collocation schemes. With this discretization, a Helmholtz problem on a domain of size 1000 λ× 1000 λ (for which N=100M) is solved in 15 minutes to 6 correct digits on a high-powered desktop.
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